Two-Lines: A Valid Alternative to the Invalid Testing of U-Shaped Relationships with Quadratic Regressions
37 Pages Posted: 22 Aug 2017 Last revised: 24 Mar 2018
Date Written: March 21, 2018
Many psychological theories predict u-shaped relationships: the effect of x is positive for low values of x, but negative for high values, or vice-versa. Despite implying merely a change of sign, hypotheses about u-shapes are tested almost exclusively via quadratic regressions. The unnecessary and arbitrary quadratic assumption can lead to a 100% false-positive rate, e.g., concluding y=log(x) is u-shaped. Estimating two regression lines, one for low and one high values of x, allows testing for a sign change without functional form assumptions. To set the breakpoint between the lines, I introduce the Robin Hood algorithm. It delivers higher power to detect u-shapes than all breakpoint alternatives considered. The paper includes simulations and re-analyses of published results.
The app: http://webstimate.org/twolines
Keywords: regression, interactoin, quadratic, u-shape, specification error
Suggested Citation: Suggested Citation